4.4.3 Monte Carlo Algorithm for the Impulse Response

Using (4.57) and the combined rejection technique developed for the secondary trajectories based on the inequalities (4.78) to (4.80), the new small-signal Monte Carlo algorithm including the Pauli exclusion principle can be formulated as follows:

1. Simulate the nonlinear Boltzmann equation until $f_{s}$ has converged.
2. Follow a main trajectory for one free flight. Store the before-scattering state in $\vec{k}_{b}$, and realize a scattering event from $\vec{k}_{b}$ to $\vec {k}_{a}$.
3. Start a trajectory $\vec{K}^{+}(t)$ from $\vec{k}_{b}$ and another trajectory $\vec{K}^{-}(t)$ from $\vec {k}_{a}$.
4. Follow both trajectories for time $T$ using the rejection scheme based on the acceptance conditions (4.78) to (4.80). At equidistant times $t_{i}$ add $A(\vec{K}^{+}(t_{i}))$ to a histogram $\alpha_{i}^{+}$ and $A(\vec{K}^{-}(t_{i}))$ to a histogram $\alpha_{i}^{-}$.
5. Continue with the second step until $N$ $\vec{k}$-points have been generated.
6. Calculate the time discrete impulse response as $\langle A \rangle_{im}(t_{i})=(E_{im}\langle\lambda\rangle/NE_{s})(\alpha_{i}^{+}-\alpha_{i}^{-})$.

This algorithm is schematically illustrated in Fig. 4.7 and its flow chart is shown in Fig. 4.8.

Figure 4.7: Schematic representation of the small-signal algorithm.

It should be noted that in a highly degenerate electron gas the main trajectory contains many self-scattering events. As in this case $\vec{k}_{a}=\vec{k}_{b}$, the corresponding two secondary trajectories will give the same contribution, that is $\alpha_{i}^{+}=\alpha_{i}^{-}$. This does not change the impulse response. Thus in order to save the computation time it is reasonable not to start trajectories $K^{+}(t)$ and $K^{-}(t)$ after a self-scattering event has occurred.

Figure 4.8: Flow chart of the small-signal algorithm.

If a self-scattering event has taken place during the evolution of the main trajectory, the main trajectory is continued until a physical scattering event has happened. Only at this moment the secondary trajectories $K^{+}(t)$ and $K^{-}(t)$ are started.

S. Smirnov: